In mathematical terms, given a smooth manifoldM{\displaystyle {\mathcal {M}}}, two discrete, freely acting, groups G1{\displaystyle G_{1}} and G2{\displaystyle G_{2}} and the worldsheetparity operator Ωp{\displaystyle \Omega _{p}} (such that Ωp:σ→2π−σ{\displaystyle \Omega _{p}:\sigma \to 2\pi -\sigma }) an orientifold is expressed as the quotient space M/(G1∪ΩG2){\displaystyle {\mathcal {M}}/(G_{1}\cup \Omega G_{2})}. If G2{\displaystyle G_{2}} is empty, then the quotient space is an orbifold. If G2{\displaystyle G_{2}} is not empty, then it is an orientifold.

Application to string theory

In string theory M{\displaystyle {\mathcal {M}}} is the compact space formed by rolling up the theory's extra dimensions, specifically a six-dimensional Calabi-Yau space. The simplest viable compact spaces are those formed by modifying a torus.

Supersymmetry breaking

The six dimensions take the form of a Calabi-Yau for reasons of partially breaking the supersymmetry of the string theory to make it more phenomenologically viable. The Type II string theories have 32 real supercharges, and compactifying on a six-dimensional torus leaves them all unbroken. Compactifying on a more general Calabi-Yau sixfold, 3/4 of the supersymmetry is removed to yield a four-dimensional theory with 8 real supercharges (N=2). To break this further to the only non-trivial phenomenologically viable supersymmetry, N=1, half of the supersymmetry generators must be projected out and this is achieved by applying the orientifold projection.

Effect on field content

A simpler alternative to using Calabi-Yaus to break to N=2 is to use an orbifold originally formed from a torus. In such cases it is simpler to examine the symmetry group associated to the space as the group is given in the definition of the space.

The orbifold group G1{\displaystyle G_{1}} is restricted to those groups which work crystallographically on the torus lattice,[1] i.e. lattice preserving. G2{\displaystyle G_{2}} is generated by an involutionσ{\displaystyle \sigma }, not to be confused with the parameter signifying position along the length of a string. The involution acts on the holomorphic 3-form Ω{\displaystyle \Omega } (again, not to be confused with the parity operator above) in different ways depending on the particular string formulation being used.[2]

The locus where the orientifold action reduces to the change of the string orientation is called the orientifold plane. The involution leaves the large dimensions of space-time unaffected and so orientifolds can have O-planes of at least dimension 3. In the case of σ(Ω)=Ω{\displaystyle \sigma (\Omega )=\Omega } it is possible that all spatial dimensions are left unchanged and O9 planes can exist. The orientifold plane in type I string theory is the spacetime-filling O9-plane.

More generally, one can consider orientifold Op-planes where the dimension p is counted in analogy with Dp-branes. O-planes and D-branes can be used within the same construction and generally carry opposite tension to one another.

However, unlike D-branes, O-planes are not dynamical. They are defined entirely by the action of the involution, not by string boundary conditions as D-branes are. Both O-planes and D-branes must be taken into account when computing tadpole constraints.

This has the result that the number of moduli parameterising the space is reduced. Since σ{\displaystyle \sigma } is an involution, it has eigenvalues ±1{\displaystyle \pm 1}. The (1,1)-form basis ωi{\displaystyle \omega _{i}}, with dimension h1,1{\displaystyle h^{1,1}} (as defined by the Hodge Diamond of the orientifold's cohomology) is written in such a way that each basis form has definite sign under σ{\displaystyle \sigma }. Since moduli Ai{\displaystyle A_{i}} are defined by J=Aiωi{\displaystyle J=A_{i}\omega _{i}} and J must transform as listed above under σ{\displaystyle \sigma }, only those moduli paired with 2-form basis elements of the correct parity under σ{\displaystyle \sigma } survive. Therefore, σ{\displaystyle \sigma } creates a splitting of the cohomology as h1,1=h+1,1+h−1,1{\displaystyle h^{1,1}=h_{+}^{1,1}+h_{-}^{1,1}} and the number of moduli used to describe the orientifold is, in general, less than the number of moduli used to describe the orbifold used to construct the orientifold.[3] It is important to note that although the orientifold projects out half of the supersymmetry generators the number of moduli it projects out can vary from space to space. In some cases h1,1=h±1,1{\displaystyle h^{1,1}=h_{\pm }^{1,1}}, in that all of the (1-1)-forms have the same parity under the orientifold projection. In such cases the way in which the different supersymmetry content enters into the moduli behaviour is through the flux dependent scalar potential the moduli experience,the N=1 case is different from the N=2 case.

A black string is a higher dimensional (D>4) generalization of a black hole in which the event horizon is topologically equivalent to S2 × S1 and spacetime is asymptotically Md−1 × S1.

Perturbations of black string solutions were found to be unstable for L (the length around S1) greater than some threshold L′. The full non-linear evolution of a black string beyond this threshold might result in a black string breaking up into separate black holes which would coalesce into a single black hole. This scenario seems unlikely because it was realized a black string could not pinch off in finite time, shrinking S2 to a point and then evolving to some Kaluza–Klein black hole. When perturbed, the black string would settle into a stable, static non-uniform black string state.

In particle physics, the hypothetical dilaton particle, and scalar field, appears in theories with extra dimensions when the volume of the compactified dimensions varies. It appears as a radion in Kaluza–Klein theory's compactifications of extra dimensions. A particle of a scalar field Φ, a scalar field that always comes with gravity, and in a dynamical field the resulting dilaton particle parallels the graviton. For comparison, in standard general relativity, Newton's constant, or equivalently the Planck mass is a constant.

In theoretical physics, a Hořava–Witten domain wall is a type of domain wall that behaves as a boundary of the eleven-dimensional spacetime in M-theory.

Petr Hořava and Edward Witten argued that the cancellation of anomalies guarantees that a supersymmetric gauge theory with the E8 gauge group propagates on this domain wall. This fact is important for various relations between M-theory and superstring theory.

In theoretical physics, an M2-brane, is a spatially extended mathematical object (brane) that appears in string theory and in related theories (e.g. M-theory, F-theory). In particular, it is a solution of eleven-dimensional supergravity which possesses a three-dimensional world volume.

In theoretical physics, an M5-brane is a brane which carries magnetic charge, and the dual under electric-magnetic duality is the M2-brane. M5-brane is analogous to the NS5-brane in string theory. In addition, it is a soliton solution to M-theory.

In physics, matrix string theory is a set of equations that describe superstring theory in a non-perturbative framework. Type IIA string theory can be shown to be equivalent to a maximally supersymmetric two-dimensional gauge theory, the gauge group of which is U(N) for a large value of N. This matrix string theory was first proposed by Luboš Motl in 1997 and later independently in a more complete paper by Robbert Dijkgraaf, Erik Verlinde, and Herman Verlinde. Another matrix string theory equivalent to Type IIB string theory was constructed in 1996 by Ishibashi, Kawai, Kitazawa and Tsuchiya. This version is known as the IKKT matrix model.

In string theory, N=2 superstring is a theory in which the worldsheet admits N=2 supersymmetry rather than N=1 supersymmetry as in the usual superstring. The target space (a term used for a generalization of space-time) is four-dimensional, but either none or two of its dimensions are time-like, i.e. it has either 4+0 or 2+2 dimensions. The spectrum consists of only one massless scalar, which describes gravitational fluctuations of self-dual gravity. The target space theory is therefore self-dual gravity, and is thought to consist no local (or propagating) degrees of freedom.

In mathematics, Ricci-flat manifolds are Riemannian manifolds whose Ricci curvature vanishes. Ricci-flat manifolds are special cases of Einstein manifolds, where the cosmological constant need not vanish.

Since Ricci curvature measures the amount by which the volume of a small geodesic ball deviates from the volume of a ball in Euclidean space, small geodesic balls will have no volume deviation, but their "shape" may vary from the shape of the standard ball in Euclidean space. For example, in a Ricci-flat manifold, a circle in Euclidean space may be deformed into an ellipse with equal area. This is due to Weyl curvature.

Ricci-flat manifolds often have restricted holonomy groups. Important cases include Calabi–Yau manifolds and hyperkähler manifolds.

In string theory, an S-brane is a hypothetical and controversial counterpart of a D-brane, which, unlike a D-brane, is localized in time. Depending on the context the "S" stands for "Strominger", "Sen", or "Space-like". The S-brane was originally proposed by Andrew Strominger in his speculative paper with Michael Gutperle, and another version of S-branes was studied by Ashoke Sen. They are thought to be extended in the space-like/temporal directions only, so that they exist at only one point in time, but are otherwise analogous to p-branes.

In theoretical physics, a string background refers to the set of classical values of quantum fields in spacetime that correspond to classical solutions of string theory. Such a background is associated with geometry that solves Einstein's equations (with higher order corrections) or their generalizations and with the values of other fields. These fields may encode the information about the shape of the hidden dimensions; the size of various electromagnetic fields and their generalizations; the values of fluxes; and the presence of additional objects such as D-branes and orientifold planes. The full physics of string theory can always be thought of as a system of infinitely many quantum fields expanded around a given string background.

In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. It describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries gravitational force. Thus string theory is a theory of quantum gravity.

String theory is a broad and varied subject that attempts to address a number of deep questions of fundamental physics. String theory has been applied to a variety of problems in black hole physics, early universe cosmology, nuclear physics, and condensed matter physics, and it has stimulated a number of major developments in pure mathematics. Because string theory potentially provides a unified description of gravity and particle physics, it is a candidate for a theory of everything, a self-contained mathematical model that describes all fundamental forces and forms of matter. Despite much work on these problems, it is not known to what extent string theory describes the real world or how much freedom the theory allows in the choice of its details.

String theory was first studied in the late 1960s as a theory of the strong nuclear force, before being abandoned in favor of quantum chromodynamics. Subsequently, it was realized that the very properties that made string theory unsuitable as a theory of nuclear physics made it a promising candidate for a quantum theory of gravity. The earliest version of string theory, bosonic string theory, incorporated only the class of particles known as bosons. It later developed into superstring theory, which posits a connection called supersymmetry between bosons and the class of particles called fermions. Five consistent versions of superstring theory were developed before it was conjectured in the mid-1990s that they were all different limiting cases of a single theory in eleven dimensions known as M-theory. In late 1997, theorists discovered an important relationship called the AdS/CFT correspondence, which relates string theory to another type of physical theory called a quantum field theory.

One of the challenges of string theory is that the full theory does not have a satisfactory definition in all circumstances. Another issue is that the theory is thought to describe an enormous landscape of possible universes, and this has complicated efforts to develop theories of particle physics based on string theory. These issues have led some in the community to criticize these approaches to physics and question the value of continued research on string theory unification.

In theoretical physics, type II string theory is a unified term that includes both type IIA strings and type IIB strings theories. Type II string theory accounts for two of the five consistent superstring theories in ten dimensions. Both theories have the maximal amount of supersymmetry — namely 32 supercharges — in ten dimensions. Both theories are based on oriented closed strings. On the worldsheet, they differ only in the choice of GSO projection.

In theoretical physics, type I string theory is one of five consistent supersymmetric string theories in ten dimensions. It is the only one whose strings are unoriented (both orientations of a string are equivalent) and which contains not only closed strings, but also open strings.

In physics, U-duality (short for unified duality) is a symmetry of string theory or M-theory combining S-duality and T-duality transformations. The term is most often met in the context of the "U-duality (symmetry) group" of M-theory as defined on a particular background space (topological manifold). This is the union of all the S-duality and T-duality available in that topology. The narrow meaning of the word "U-duality" is one of those dualities that can be classified neither as an S-duality, nor as a T-duality - a transformation that exchanges a large geometry of one theory with the strong coupling of another theory, for example.

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